JEE Main & Advanced Mathematics Definite Integration Question Bank Area Bounded by Region, Volume and Surface Area of Solids of Revolution

  • question_answer The area bounded by the curves \[y={{\log }_{e}}x\] and \[y={{({{\log }_{e}}x)}^{2}}\] is                                               [RPET 2000]

    A)            \[3-e\]                                    

    B)            \[e-3\]

    C)                     \[\frac{1}{2}(3-e)\]   

    D)                     \[\frac{1}{2}(e-3)\]

    Correct Answer: A

    Solution :

               Required area \[=\int_{1}^{e}{[\log x-{{(\log x)}^{2}}]}\,dx\]            \[A=\int_{1}^{e}{\,\log x\,dx}-\int_{1}^{e}{{{(\log x)}^{2}}dx}\]               \[=[x\log x-x]\,_{1}^{e}-[x{{(\log x)}^{2}}-2x\log x+2x]\,_{1}^{e}\]               \[=[e-e-(-1)]-[e{{(1)}^{2}}-2e+2e-(2)]\]               \[=(1)-(e-2)\]\[=3-e\].

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